Singer's cohomological transfer. The next contribution of the paper is to apply Theorem REF  into the study of the behavior of the fifth transfer homomorphism of W. Singer [1]}. It may need to be recalled that the group \(GL_t\)  acts regularly on \(P_t\)  by matrix substitution. Further, the two actions of \(\mathcal {A}_2\)  and \(GL_t\)  upon \(P_t\)  commute with each other; hence there is an inherited action of \(GL_t\)  on \(QP_t = \mathbb {Z}_2\otimes _{\mathcal {A}_2}P_t.\)  Thus, the hit problem becomes important in studying the modular representation of \(GL_t.\)  This is of interest since the Singer cohomological "transfer" [1]} relates \((QP_t)_n\)  to \(H^{t, t+n}(\mathcal {A}_2, \mathbb {Z}_2) = {\rm Ext}_{\mathcal {A}_2}^{t, t+n}(\mathbb {Z}_2, \mathbb {Z}_2),\)  the \(\mathbb {Z}_2\) -cohomology groups of \(\mathcal {A}_2,\)  and thus to the stable homotopy groups of spheres. Singer's transfer is established as follows. Let us recall first that if \(\mathcal {M}\)  the category of graded left \(\mathcal {A}_2\) -modules and degree zero of \(\mathcal {A}_2\) -linear maps, then for each integer \(t,\)  the supension functor \(S^{t}: \mathcal {M}\rightarrow \mathcal {M}\) is defined by \((S^{t}U)^{n} := U^{n-t}\) where \(n\in \mathbb {Z}\). The action of \(\mathcal {A}_2\)  on \(S^{t}U\)  is given by \(\theta (S^{t}u) = S^{t}(\theta u),\) for all \(u\in U\)  and \(\theta \in \mathcal {A}_2.\)  Next, consider the polynomial ring \(P_1 = \mathbb {Z}_2[x]\) with \(\deg (x) = 1,\)  the canonical \(\mathcal {A}_2\) -action on \(P_1\)  is extended to an \(\mathcal {A}_2\) -action on the ring of finite Laurent series \(\mathbb {Z}_2[x_1, x_1^{-1}]\) . Then, there exists an \(\mathcal {A}_2\) -submodule \(\overline{P} = \langle \lbrace x_1^{t}|\ t\ge -1\rbrace \rangle \) of \(\mathbb {Z}_2[x_1, x_1^{-1}].\)  One has a short-exact sequence \( 0\rightarrow P_1\xrightarrow{} \overline{P}\xrightarrow{} S^{-1}\mathbb {Z}_2,\)  where \(q\)  is the inclusion and \(\pi \)  is given by \(\pi (x_1^{t}) = 0\)  if \(t\ne -1\)  and \(\pi (x_1^{-1}) = 1.\)  Writing \(e_1\)  for the corresponding element in \({\rm Ext}_{\mathcal {A}_2}^1(S^{-1}\mathbb {Z}_2, P_1).\)  Basing the cross, the Yoneda and the cap products in co(homology) with \(\mathbb {Z}_2\) -coeffcients, we have a homomorphism
\( \begin{array}{ll}Tr^{\mathcal {A}_2}_t: {\rm Tor}_t^{\mathcal {A}_2}(\mathbb {Z}_2, S^{-t}\mathbb {Z}_2) &\longrightarrow {\rm Tor}_0^{\mathcal {A}_2}(\mathbb {Z}_2, P_t) = QP_t\\\hspace{56.9055pt} z&\longmapsto e_t\cap z,\end{array}\) 
